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Signals, modulation, noise, multiplexing and digital communication basics.
| Parameter | Formula | Unit |
|---|---|---|
| Shannon Capacity | C = B log₂(1 + SNR) | bps |
| Nyquist Rate (Noiseless) | C_max = 2B log₂(L) | bps (L = signal levels) |
| Signal-to-Noise Ratio | SNR = P_signal / P_noise | ratio or dB |
| SNR in dB | SNR(dB) = 10 log₁₀(SNR) | dB |
| Bandwidth-Time Product | B × T ≥ 1/2 | Hz·s |
| Energy per Bit | E_b = P_signal × T_b | Joules |
| E_b/N₀ | E_b/N₀ = (S/N) × (B/R_b) | ratio or dB |
| Transform | Time → Freq | Key Property |
|---|---|---|
| Fourier Series | Periodic → Discrete spectrum | f₀ = 1/T; harmonics at nf₀ |
| Fourier Transform | Aperiodic → Continuous | X(f) = ∫x(t)e^(-j2πft) dt |
| Parseval's Theorem | ∫|x(t)|²dt = ∫|X(f)|²df | Energy conservation |
| Convolution Theorem | x(t)*h(t) ↔ X(f)·H(f) | Time convolution ↔ freq multiplication |
┌─────────────────────────────────────────────────────────────────┐
│ SHANNON-HARTLEY THEOREM │
├─────────────────────────────────────────────────────────────────┤
│ │
│ C = B × log₂(1 + SNR) │
│ │
│ Where: │
│ C = Maximum channel capacity (bits/second) │
│ B = Channel bandwidth (Hz) │
│ SNR = Signal-to-Noise Ratio (linear, not dB!) │
│ │
│ WORKED EXAMPLES: │
│ ──────────────── │
│ Example 1: Telephone channel │
│ B = 3000 Hz, SNR = 30 dB = 1000 │
│ C = 3000 × log₂(1001) = 3000 × 9.97 ≈ 29,900 bps ≈ 30 kbps│
│ Note: 56k modems approach this via advanced coding │
│ │
│ Example 2: WiFi (802.11n) │
│ B = 20 MHz, SNR = 20 dB = 100 │
│ C = 20×10⁶ × log₂(101) = 20×10⁶ × 6.66 ≈ 133 Mbps │
│ With MIMO and coding: up to 150 Mbps per spatial stream │
│ │
│ Example 3: Deep Space (low SNR) │
│ B = 10 MHz, SNR = 0.1 (-10 dB) │
│ C = 10×10⁶ × log₂(1.1) = 10×10⁶ × 0.137 ≈ 1.37 Mbps │
│ Trade bandwidth for power: increase B to compensate for SNR │
│ │
│ KEY INSIGHTS: │
│ • To double C: either double B OR increase SNR by 3 dB │
│ • Error-free communication only if R_b ≤ C │
│ • SNR < 1 (negative dB): still possible if B is large enough! │
│ • For binary: C → ∞ as B → ∞ (spread spectrum principle) │
└─────────────────────────────────────────────────────────────────┘┌─────────────────────────────────────────────────────────────────┐
│ DECIBEL CALCULATIONS │
├─────────────────────────────────────────────────────────────────┤
│ │
│ POWER RATIO: dB = 10 log₁₀(P₂/P₁) │
│ VOLTAGE RATIO: dB = 20 log₁₀(V₂/V₁) │
│ CURRENT RATIO: dB = 20 log₁₀(I₂/I₁) │
│ │
│ COMMON dB VALUES TO MEMORIZE: │
│ 1 dB = ×1.26 (power) = ×1.12 (voltage) │
│ 3 dB = ×2.00 (power) = ×1.41 (voltage) │
│ 6 dB = ×4.00 (power) = ×2.00 (voltage) │
│ 10 dB = ×10.0 (power) = ×3.16 (voltage) │
│ 20 dB = ×100 (power) = ×10.0 (voltage) │
│ 30 dB = ×1000 (power) = ×31.6 (voltage) │
│ │
│ ABSOLUTE POWER UNITS: │
│ dBm = 10 log₁₀(P / 1mW) │
│ dBW = 10 log₁₀(P / 1W) │
│ dBm to dBW: subtract 30 (1W = 1000mW = +30 dBm) │
│ │
│ CONVERSION EXAMPLE: │
│ 0 dBm = 1 mW | +10 dBm = 10 mW │
│ +20 dBm = 100 mW | +30 dBm = 1 W │
│ -10 dBm = 0.1 mW | -30 dBm = 1 µW │
└─────────────────────────────────────────────────────────────────┘L levels require log₂(L) bits per symbol, so increasing levels trades noise immunity for data rate.┌─────────────────────────────────────────────────────────────────┐
│ AMPLITUDE MODULATION — TYPES & FORMULAS │
├─────────────────────────────────────────────────────────────────┤
│ │
│ STANDARD AM (DSB + Carrier): │
│ s(t) = A_c [1 + m cos(2πf_m t)] cos(2πf_c t) │
│ where m = modulation index (0 < m ≤ 1) │
│ m = V_m / V_c (ratio of modulating to carrier amplitude) │
│ │
│ Spectrum: Carrier at f_c, sidebands at f_c ± f_m │
│ BW = 2 f_m (two sidebands, each of width f_m) │
│ │
│ POWER IN STANDARD AM: │
│ P_total = P_c × (1 + m²/2) │
│ P_carrier = P_c │
│ P_sidebands = P_c × m²/2 │
│ Max efficiency (m=1): η = m²/(2+m²) = 1/3 = 33.3% │
│ ⚠️ 66.7% of power wasted in carrier! │
│ │
│ DSB-SC (Double Sideband — Suppressed Carrier): │
│ s(t) = A_c × m(t) × cos(2πf_c t) │
│ No carrier component! │
│ BW = 2 f_m (same as AM) │
│ Efficiency = 100% (all power in sidebands) │
│ Problem: coherent detection needed (receiver needs phase ref)│
│ Coherent detector: multiply by cos(2πf_c t) + LPF │
│ │
│ SSB (Single Sideband): │
│ USB: s(t) = (A_c/2) m(t) cos(2πf_c t) - (A_c/2) m̂(t)sin │
│ LSB: s(t) = (A_c/2) m(t) cos(2πf_c t) + (A_c/2) m̂(t)sin │
│ where m̂(t) = Hilbert transform of m(t) │
│ BW = f_m (half of DSB!) │
│ Efficiency = 100%, minimum bandwidth │
│ Used in: HF radio, military, amateur radio │
│ Generation: Filter method or Phase-shift method │
│ │
│ VSB (Vestigial Sideband): │
│ Compromise between SSB and DSB │
│ Keeps vestige of one sideband + full other sideband │
│ BW: f_m < BW < 2f_m │
│ Used in: TV video transmission │
└─────────────────────────────────────────────────────────────────┘| Type | BW | Power Efficiency | Carrier | Detection |
|---|---|---|---|---|
| Standard AM | 2f_m | 33.3% (max) | Present | Envelope detector (simple) |
| DSB-SC | 2f_m | 100% | Suppressed | Coherent/synchronous detector |
| SSB | f_m | 100% | Suppressed | Coherent detector or carrier reinsertion |
| VSB | f_m + vestige | ~100% | Suppressed | Envelope + special filter |
MODULATION INDEX (m):
m = (V_max - V_min) / (V_max + V_min)
m < 1 → Under-modulation (safe, inefficient)
m = 1 → 100% modulation (optimal AM)
m > 1 → Over-modulation (DISTORTION!)
Total harmonic distortion when m > 1:
D = √(Σ Vₙ²) / V₁
ENVELOPE DETECTOR CONDITIONS:
• RC >> 1/f_c (carrier tracking)
• RC << 1/f_m (modulation tracking)
• Rule: 1/f_c << RC << 1/f_m
• Diode: conducts on +peak, C holds voltage during -halfenvelope detectoronly works for standard AM with m ≤ 1 — it's the simplest and cheapest receiver design (diode + RC filter).┌─────────────────────────────────────────────────────────────────┐
│ FREQUENCY & PHASE MODULATION │
├─────────────────────────────────────────────────────────────────┤
│ │
│ FM SIGNAL: │
│ s(t) = A_c cos[2πf_c t + β sin(2πf_m t)] │
│ where β = Δf / f_m = modulation index │
│ │
│ PM SIGNAL: │
│ s(t) = A_c cos[2πf_c t + k_p m(t)] │
│ where k_p = phase deviation constant (rad/V) │
│ │
│ KEY RELATIONSHIP: FM ↔ PM │
│ • FM: phase deviation proportional to integral of message │
│ • PM: frequency deviation proportional to derivative of msg │
│ • PM from FM: differentiate m(t) before FM modulator │
│ • FM from PM: integrate m(t) before PM modulator │
│ │
│ CARSON'S RULE (Bandwidth): │
│ BW ≈ 2(Δf + f_m) = 2f_m(β + 1) │
│ For narrowband FM (β << 1): BW ≈ 2f_m (like AM) │
│ For wideband FM (β >> 1): BW ≈ 2Δf = 2βf_m │
│ │
│ FM PARAMETERS: │
│ Δf = k_f × A_m (max frequency deviation) │
│ β = Δf / f_m (modulation index) │
│ k_f = frequency sensitivity (Hz/V) │
│ │
│ FM POWER: │
│ P_FM = A_c² / 2 = constant! │
│ Power in FM is always constant (envelope is constant) │
│ Power distributes among carrier + many sidebands │
│ (Bessel functions determine sideband amplitudes) │
│ │
│ FM vs AM COMPARISON: │
│ ┌────────────────┬──────────────────┬──────────────────┐ │
│ │ Feature │ AM │ FM │ │
│ ├────────────────┼──────────────────┼──────────────────┤ │
│ │ BW │ 2f_m │ 2(Δf + f_m) │ │
│ │ SNR improvement│ None │ 3β² (wideband) │ │
│ │ Capture effect │ No │ Yes (stronger wins│ │
│ │ Envelope │ Varies │ Constant │ │
│ │ Power │ Varies with m │ Constant │ │
│ │ Complexity │ Simple │ More complex │ │
│ └────────────────┴──────────────────┴──────────────────┘ │
└─────────────────────────────────────────────────────────────────┘| Device | Type | Principle | Notes |
|---|---|---|---|
| Varactor Diode | Direct FM | C varies with voltage → changes VCO frequency | Simple, requires AFC for stability |
| Voltage Controlled Oscillator | Direct FM | Input voltage controls frequency | Used in PLL-based FM |
| Armstrong Method | Indirect FM (PM→FM) | Narrowband PM + frequency multipliers | Better frequency stability (crystal reference) |
| Foster-Seeley / Ratio Detector | FM demod | Phase shift between tuned circuits → voltage | Classic analog FM detector |
| PLL (Phase-Locked Loop) | FM demod | VCO tracks frequency → control voltage = m(t) | Modern, excellent performance |
| Quadrature Detector | FM demod | 90° phase shift → amplitude variation → envelope detect | IC implementation (MC3361) |
| Parameter | Pre-Emphasis (TX) | De-Emphasis (RX) |
|---|---|---|
| Purpose | Boost high frequencies before modulation | Attenuate high frequencies after demod |
| Circuit | HPF: f_c = 1/(2πR₁C) | LPF: f_c = 1/(2πRC) |
| Time Constant | τ = 75 µs (US/FM broadcast) | τ = 75 µs (US/FM broadcast) |
| SNR Improvement | — | Up to 13 dB improvement! |
| Transfer Function | H(f) = 1 + jf/f₁ | H(f) = 1 / (1 + jf/f₁) |
BW ≈ 2(Δf + f_m). The capture effect means that in FM, the stronger signal completely suppresses the weaker one (no co-channel interference). For commercial FM broadcast (88–108 MHz), Δf = 75 kHz, f_m(max) = 15 kHz, so BW ≈ 180 kHz per station.| Scheme | Symbol | Bandwidth | BER Formula | Spectral Efficiency |
|---|---|---|---|---|
| ASK/OOK | Amplitude | 2R_b | Q(√(E_b/2N₀)) | 0.5 bps/Hz |
| BFSK | Frequency | 2Δf + 2R_b | Q(√(E_b/N₀)) | ~0.5 bps/Hz |
| BPSK | Phase (0, π) | 2R_b | Q(√(2E_b/N₀)) | 1 bps/Hz |
| QPSK | 4-phase (±45°, ±135°) | R_b | Q(√(2E_b/N₀)) (per bit) | 2 bps/Hz |
| 8-PSK | 8-phase | R_b/1.5 | Worse than QPSK | 3 bps/Hz |
| 16-QAM | 16 amplitude+phase | R_b/2 | Worse than QPSK | 4 bps/Hz |
| 64-QAM | 64 amplitude+phase | R_b/3 | Worse still | 6 bps/Hz |
| 256-QAM | 256 amplitude+phase | R_b/4 | Needs high SNR | 8 bps/Hz |
┌─────────────────────────────────────────────────────────────────┐
│ DIGITAL MODULATION — CONSTELLATION DIAGRAMS │
├─────────────────────────────────────────────────────────────────┤
│ │
│ BPSK (Binary Phase Shift Keying): │
│ Symbols: (1,0) → +√E_s and (-1,0) → -√E_s │
│ 1 bit per symbol | d_min = 2√E_s │
│ Simplest, most robust, used in GPS, satellite downlink │
│ │
│ QPSK (Quadrature Phase Shift Keying): │
│ Symbols: (±1, ±1) × √(E_s/2) │
│ 2 bits per symbol (I and Q channels) │
│ d_min = √(2E_s) │
│ Same BER as BPSK per bit, but double the data rate! │
│ Used in: DVB-S, Wi-Fi, CDMA │
│ │
│ QAM (Quadrature Amplitude Modulation): │
│ Combines amplitude + phase variations │
│ 16-QAM: 4×4 grid, 4 bits/symbol │
│ 64-QAM: 8×8 grid, 6 bits/symbol │
│ Trade-off: Higher order → more bits/symbol → needs more SNR │
│ Used in: WiFi (OFDM), LTE, Cable modems, DVB-C │
│ │
│ M-ARY RELATIONSHIPS: │
│ k bits per symbol → M = 2^k symbols │
│ Symbol rate: R_s = R_b / k │
│ Bandwidth: BW ≈ R_s = R_b / k │
│ Energy per symbol: E_s = k × E_b │
│ E_b/N₀ needed increases with M (for same BER) │
│ │
│ π/4-DQPSK: Differential QPSK (no carrier recovery needed) │
│ Transitions: ±45° and ±135° (no 180° transitions) │
│ Used in: GSM, North American TDMA (IS-136), TETRA │
└─────────────────────────────────────────────────────────────────┘| Modulation | BER Approx. | E_b/N₀ for BER=10⁻⁵ |
|---|---|---|
| OOK (ASK) | Q(√(E_b/2N₀)) | ~14.2 dB |
| BFSK (Coherent) | Q(√(E_b/N₀)) | ~12.6 dB |
| BFSK (Non-coherent) | ½ exp(-E_b/2N₀) | ~13.5 dB |
| BPSK | Q(√(2E_b/N₀)) | ~9.6 dB |
| QPSK | ≈ Q(√(2E_b/N₀)) per bit | ~9.6 dB |
| 16-QAM | ≈ 3Q(√(4E_b/5N₀)) | ~14.5 dB |
| 64-QAM | Worse than 16-QAM | ~20 dB |
Q-FUNCTION: Q(x) = ½ erfc(x/√2)
BER for BPSK:
P_b = Q(√(2E_b/N₀))
BER for BFSK:
P_b = Q(√(E_b/N₀))
BER for OOK:
P_b = Q(√(E_b/2N₀))
BER for QPSK (approximate):
P_b ≈ Q(√(2E_b/N₀))
RULE: BPSK needs 3 dB less E_b/N₀ than BFSK for same BER
(because √2 factor in argument of Q-function)adaptive modulation — switching between QPSK, 16-QAM, and 64-QAM depending on channel conditions (e.g., LTE, WiFi).| Parameter | Description |
|---|---|
| Principle | Divide total bandwidth into non-overlapping frequency bands |
| Guard Band | Small frequency gap between channels to prevent interference |
| Total BW | N × B_ch + (N-1) × B_guard |
| Modulation | Each channel modulates a different carrier frequency |
| Used In | FM radio, TV broadcasting, analog telephone (12 voice channels/group) |
| ADC in FDM | Each channel needs its own modulator/demodulator pair |
| Parameter | Description |
|---|---|
| Principle | Each user gets entire bandwidth for a time slot |
| Frame | One complete cycle of all user slots + sync bits |
| Frame Rate | f_s = sampling rate (e.g., 8000 Hz for voice) |
| Bit Duration | T_b = T_slot / n_bits_per_channel |
| Efficiency | η = N × T_slot / T_frame (excluding overhead) |
| Used In | T1/E1 digital telephone, GSM, SONET/SDH |
┌─────────────────────────────────────────────────────────────────┐
│ MULTIPLEXING — PRACTICAL SYSTEMS │
├─────────────────────────────────────────────────────────────────┤
│ │
│ T1 CARRIER (North America): │
│ 24 voice channels, 8-bit PCM, 8 kHz sampling │
│ Bit rate = 24 × 8 × 8000 + 1 framing bit × 8000 │
│ = 1.544 Mbps (DS-1) │
│ Frame: 193 bits (24 × 8 + 1), 8000 frames/sec │
│ │
│ E1 CARRIER (Europe): │
│ 30 voice channels + 1 signaling + 1 sync = 32 channels │
│ Bit rate = 32 × 8 × 8000 = 2.048 Mbps │
│ Frame: 256 bits, 8000 frames/sec │
│ │
│ HIERARCHY: │
│ DS-1 (T1): 1.544 Mbps (24 channels) │
│ DS-2: 6.312 Mbps (96 channels = 4 × T1) │
│ DS-3: 44.736 Mbps (672 channels = 28 × T1) │
│ OC-1 (SONET): 51.84 Mbps │
│ OC-3 (SONET): 155.52 Mbps │
│ OC-12 (SONET): 622.08 Mbps │
│ OC-48 (SONET): 2.488 Gbps │
│ │
│ CDMA (Code Division Multiple Access): │
│ • All users share same frequency and time simultaneously │
│ • Each user assigned unique spreading code (PN sequence) │
│ • Signal multiplied by code → spread over wide BW │
│ • Receiver despreads using same code │
│ • Processing gain: G_p = BW_spread / BW_data │
│ • Near-far problem: needs power control │
│ • Used in: CDMA2000, WCDMA (3G), GPS │
│ │
│ OFDMA (Orthogonal Frequency Division Multiple Access): │
│ • Subcarriers are orthogonal (no guard bands needed) │
│ • Each user gets subset of subcarriers │
│ • Used in: 4G LTE, 5G NR, WiFi 6 │
│ • Very efficient spectrum utilization │
└─────────────────────────────────────────────────────────────────┘OFDMA, which is essentially FDM but with orthogonal subcarriers (no guard bands needed between subcarriers).| Antenna | Type | Gain (dBi) | BW | Application |
|---|---|---|---|---|
| Isotropic | Reference (theoretical) | 0 | — | Reference for gain calculation |
| Half-Wave Dipole | Wire, λ/2 | 2.15 | ~3% narrow | TV, FM radio, general purpose |
| Quarter-Wave Monopole | Wire, λ/4 + ground plane | 5.15 | ~10% | Car radios, mobile phones |
| Yagi-Uda | Directional array | 10–15 | Narrow | TV reception, point-to-point |
| Patch (Microstrip) | Flat, λ/2 × λ/2 | 6–9 | Moderate | GPS, WiFi, satellite, phased arrays |
| Horn | Waveguide flared | 10–20 | Wide | Satellite, radar, feed for dishes |
| Parabolic Dish | Reflector | 20–50+ | Moderate | Satellite TV, deep space, radio astronomy |
| Phased Array | Multiple elements | Variable | Wide | Radar, 5G base stations |
┌─────────────────────────────────────────────────────────────────┐
│ ANTENNA — KEY PARAMETERS & FORMULAS │
├─────────────────────────────────────────────────────────────────┤
│ │
│ DIRECTIVITY (D): │
│ D = 4π × P_rad(max) / P_rad(total) │
│ D_dBi = 10 log₁₀(D) │
│ For half-wave dipole: D = 1.64 (2.15 dBi) │
│ │
│ GAIN (G): │
│ G = η × D (η = efficiency, typically 0.5–0.9) │
│ G = 4π A_e / λ² (where A_e = effective aperture) │
│ For isotropic: G = 0 dBi │
│ For dipole: G ≈ 2.15 dBi │
│ │
│ EFFECTIVE APERTURE: │
│ A_e = Gλ² / (4π) │
│ For half-wave dipole: A_e ≈ 0.13λ² │
│ For parabolic dish: A_e = η × π(D/2)² │
│ │
│ FRIIS TRANSMISSION EQUATION: │
│ P_r = (P_t × G_t × G_r × λ²) / ((4πR)²) │
│ │
│ In dB: │
│ P_r(dBm) = P_t(dBm) + G_t(dBi) + G_r(dBi) │
│ - 20log₁₀(4πR/λ) │
│ │
│ ANTENNA FACTOR (EMI measurements): │
│ AF = E / V = 9.73 / (λ × √G)) [1/m] │
│ │
│ RADIATION RESISTANCE (half-wave dipole): │
│ R_rad = 73 Ω (half-wave dipole) │
│ R_rad = 36.5 Ω (quarter-wave monopole, above ground plane) │
│ │
│ BANDWIDTH: │
│ BW = f_high - f_low │
│ Fractional BW = (f_high - f_low) / f_center │
│ Narrowband: < 1% | Wideband: 1–10% | Ultra-wide: > 10% │
└─────────────────────────────────────────────────────────────────┘dBi = dBd + 2.15). A parabolic dish's gain increases with the square of diameter: G = η(πD/λ)².| Noise Type | Source | Formula | Notes |
|---|---|---|---|
| Thermal (Johnson) | Resistor, conductor | P_n = kTB | White noise, flat spectrum |
| Shot Noise | Semiconductor junction | I_n² = 2qIΔf | Random carrier crossing |
| Flicker (1/f) | Semiconductor surface | S(f) ∝ 1/f | Dominates at low frequency |
| White Noise | Thermal, shot (combined) | PSD = N₀/2 | Flat across all frequencies |
| Quantization | ADC sampling | P_q = Δ²/12 | Δ = step size = V_ref/2ⁿ |
| Parameter | Formula | Description |
|---|---|---|
| Noise Figure (F) | F = (SNR)_in / (SNR)_out | Degradation of SNR by system (linear) |
| Noise Figure (dB) | NF = 10 log₁₀(F) | Expressed in dB (always ≥ 0 dB) |
| Noise Temperature | T_e = T₀(F-1) | Equivalent input noise temperature (K) |
| Cascaded NF | F = F₁ + (F₂-1)/G₁ + (F₃-1)/(G₁G₂) + ... | Friis formula — 1st stage dominates! |
┌─────────────────────────────────────────────────────────────────┐
│ NOISE — IMPORTANT CALCULATIONS │
├─────────────────────────────────────────────────────────────────┤
│ │
│ THERMAL NOISE POWER: │
│ P_n = kTB │
│ k = 1.38 × 10⁻²³ J/K (Boltzmann constant) │
│ T = 290 K (room temperature, T₀) │
│ B = bandwidth (Hz) │
│ │
│ P_n @ 290K per Hz = kT₀ = 1.38e-23 × 290 = 4.0 × 10⁻²¹ W/Hz│
│ P_n in dBm/Hz = 10 log₁₀(4e-21/1e-3) = -174 dBm/Hz │
│ ⚠️ MEMORIZE: -174 dBm/Hz at room temperature │
│ │
│ For B = 1 MHz: P_n = -174 + 10log₁₀(10⁶) = -174 + 60 │
│ = -114 dBm │
│ │
│ CASCaded NOISE FIGURE (FRIIS): │
│ F_total = F₁ + (F₂-1)/G₁ + (F₃-1)/(G₁G₂) + ... │
│ │
│ IMPLICATION: The FIRST stage dominates! │
│ → LNA (Low Noise Amplifier) must be FIRST component │
│ → High G₁ reduces contribution of subsequent stages │
│ │
│ EXAMPLE: │
│ Stage 1: LNA, F₁ = 2 dB (1.58), G₁ = 20 dB (100) │
│ Stage 2: Mixer, F₂ = 10 dB (10), G₂ = -6 dB (0.25) │
│ Stage 3: IF Amp, F₃ = 6 dB (4), G₃ = 30 dB (1000) │
│ │
│ F = 1.58 + (10-1)/100 + (4-1)/(100×0.25) │
│ F = 1.58 + 0.09 + 0.12 = 1.79 │
│ NF = 10 log₁₀(1.79) = 2.53 dB │
│ ⚠️ Even with noisy mixer (10 dB) and IF amp (6 dB), │
│ total NF is only 2.53 dB because LNA has high gain! │
│ │
│ BIT ERROR RATE (BER) RELATIONSHIPS: │
│ P_e = Q(√(2E_b/N₀)) for BPSK │
│ E_b/N₀ = (S/N) × (B/R_b) = SNR × (B/R_b) │
│ To find BER: convert E_b/N₀ to linear, compute Q() │
└─────────────────────────────────────────────────────────────────┘10 log₁₀(B) to get the noise power. The Friis noise formula shows that the first stage dominates the overall noise figure — this is why an LNA is always the first component in a receiver chain. Higher LNA gain reduces the noise contribution of all subsequent stages.┌─────────────────────────────────────────────────────────────────┐
│ TRANSMISSION LINE — PARAMETERS & EQUATIONS │
├─────────────────────────────────────────────────────────────────┤
│ │
│ DISTRIBUTED PARAMETERS (per unit length): │
│ R = resistance (Ω/m) │
│ L = inductance (H/m) │
│ G = conductance (S/m) │
│ C = capacitance (F/m) │
│ │
│ CHARACTERISTIC IMPEDANCE: │
│ Z₀ = √((R + jωL) / (G + jωC)) │
│ Lossless (R=G=0): Z₀ = √(L/C) │
│ For coax: Z₀ = (138/√ε_r) × log₁₀(D/d) │
│ For microstrip: Z₀ ≈ 87/√(ε_r+1.41) × ln(5.98h/(0.8w+t)) │
│ │
│ COMMON Z₀ VALUES: │
│ Coaxial cable: 50 Ω, 75 Ω │
│ Twin-lead: 300 Ω │
│ Microstrip: 50 Ω (typical) │
│ Ethernet (Cat5): 100 Ω │
│ USB: 90 Ω differential │
│ │
│ PROPAGATION CONSTANT: │
│ γ = α + jβ = √((R+jωL)(G+jωC)) │
│ α = attenuation constant (Np/m) │
│ β = phase constant (rad/m) │
│ Lossless: α = 0, β = ω√(LC) │
│ Phase velocity: v_p = ω/β = 1/√(LC) │
│ For lossless: v_p = c/√(ε_r) │
│ │
│ REFLECTION COEFFICIENT: │
│ Γ = (Z_L - Z₀) / (Z_L + Z₀) │
│ |Γ| ≤ 1 (passive loads) │
│ Short circuit: Γ = -1 (Z_L = 0) │
│ Open circuit: Γ = +1 (Z_L = ∞) │
│ Matched load: Γ = 0 (Z_L = Z₀) │
│ │
│ VSWR (Voltage Standing Wave Ratio): │
│ VSWR = (1 + |Γ|) / (1 - |Γ|) │
│ Matched: VSWR = 1 | Open/Short: VSWR = ∞ │
│ Good match: VSWR < 1.5 | Excellent: VSWR < 1.1 │
│ │
│ RETURN LOSS: │
│ RL = -20 log₁₀|Γ| (dB) │
│ Matched: RL = ∞ dB | Short: RL = 0 dB │
│ Good: RL > 15 dB | Excellent: RL > 25 dB │
│ │
│ INPUT IMPEDANCE (line of length l): │
│ Z_in = Z₀ × (Z_L + jZ₀ tan(βl)) / (Z₀ + jZ_L tan(βl)) │
│ λ/4 line: Z_in = Z₀²/Z_L (impedance inverter!) │
│ λ/2 line: Z_in = Z_L (repeats, regardless of Z₀) │
│ λ/8 line: Z_in = Z₀(Z₀ + jZ_L)/(Z_L + jZ₀) │
└─────────────────────────────────────────────────────────────────┘| Feature | Description |
|---|---|
| Center | Γ = 0 (perfect match, Z_L = Z₀) |
| Right edge | Γ = +1 (open circuit, r → ∞) |
| Left edge | Γ = -1 (short circuit, r = 0) |
| Horizontal axis | Pure resistance (X = 0) |
| Upper half | Inductive reactance (+jX) |
| Lower half | Capacitive reactance (-jX) |
| Constant-r circles | Resistance circles through center |
| Constant-x arcs | Reactance arcs tangent to right edge |
| Wavelength toward generator | Clockwise rotation on chart |
| VSWR circle | Circle centered at origin, radius = |Γ| |
| Method | Principle | BW | Application |
|---|---|---|---|
| Single Stub (shunt) | Parallel stub at distance d from load | Narrow | Coax, microstrip |
| Single Stub (series) | Series stub at distance d from load | Narrow | Microstrip |
| Double Stub | Two stubs at fixed spacing | Narrow | When d cannot be adjusted |
| Quarter-Wave Transformer | Z_T = √(Z₀ × Z_L) | Narrow | Real impedances only |
| Multi-section (Binomial) | Gradual impedance steps | Wide | Broadband matching |
| Tapered Line | Continuously varying Z₀ | Widest | Antenna feeds, broadband |
| Lumped Element (L-network) | Series L + shunt C (or vice versa) | Narrow | Low frequency, PCB |
λ/4 transformer is an impedance inverter: Z_in = Z₀²/Z_L — it can match any real impedance. The Smith chart is a graphical tool where every point represents a unique normalized impedance z = Z/Z₀ and corresponding reflection coefficient Γ.